show that any positive odd integer is of the form 4q+1 or 4q+3 ,where q is some integer.

Solution:  Let a be any odd positive integer and b=4  By division Lemma there exist integer q and r such that

a = 4q + r , Where

$\\ \Rightarrow 0\leqslant r< 4$

⇒    a = 4q or ,a =4q+1 or , a = 4q +2 or , a =4q+3 ⇒ r = 0,1,2,3

⇒ a=4q+1  or , a= 4q +3

Hence any odd integer is of the form  4q+1 or , 4q +3

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